Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Trigonometric Equations — solving sin θ = k, cos θ = k and tan θ = k and finding every solution in a given range — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise Trigonometric Equations in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Trigonometric Equations revision-guide angle, while Tutopiya’s Trigonometric Equations subtopic page owns the learning resource and the free Trigonometric Equations quiz owns the practice.

Trigonometric Equations appear in the Trigonometry unit of Cambridge IGCSE Mathematics (0580/0607) whenever examiners ask you to find angles that satisfy an equation such as 2 sin x = 1 or 3 cos θ + 1 = 0. The method is not difficult — isolate the trig ratio, use your calculator or the graph, then list every solution in the stated interval. This guide explains exactly what the subtopic covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• Trigonometric equations ask you to find angle(s) that make a trig expression true — always check the given range (e.g. 0° ≤ x ≤ 360°).
• Isolate sin, cos or tan first, then use sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator for the principal value.
• Use the CAST diagram or symmetry of trig graphs to find all solutions — not just the first one the calculator gives.
• Extended papers may use quadratic trig equations (e.g. 2 sin² x − sin x − 1 = 0) — factorise like a normal quadratic.

What are Trigonometric Equations in Cambridge IGCSE Maths?

Trigonometric Equations are equations where the unknown appears inside a sine, cosine or tangent function. In Cambridge IGCSE Mathematics you solve them by rearranging to the form sin x = k, cos x = k or tan x = k, finding one solution with the inverse function, then using graph symmetry to write down every angle in the required interval. Questions often specify 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π radians.

You can read the full explanation, worked examples and notes on Tutopiya’s Trigonometric Equations subtopic page before you attempt questions.

The core ideas you must master

These four ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Idea	What it means	How the exam uses it
Principal value	First angle from calculator inverse	”Write down one value of x”
CAST diagram	Which quadrants sin, cos, tan are positive	”Find all values of x, 0° ≤ x ≤ 360°“
Quadratic trig	Substitute u = sin x or cos x, then factorise	”Solve 2 cos² x − cos x − 1 = 0”
Radians	Solutions in terms of π	”Give your answers in radians”

How to solve a trigonometric equation — step by step

The safest method works for linear equations such as 3 sin x = 2.

1. Rearrange so sin x (or cos x or tan x) is alone on one side: sin x = 2/3.
2. Use the inverse function on your calculator: x = sin⁻¹(2/3) ≈ 41.8° (check DEG/RAD mode).
3. Sketch the CAST diagram or think about symmetry: sin is positive in quadrants I and II.
4. Find the second solution: 180° − 41.8° = 138.2°.
5. List every answer in the given range and round only if the question asks.

Once you have worked through a few, test yourself with the free Trigonometric Equations quiz — it tells you fast whether the method has actually stuck.

Linear vs quadratic trig equations: which approach does the question want?

Students lose marks by giving only one solution or by treating cos² x like cos x. Use the equation type to decide.

Equation type	What to do	Typical signal words
Linear (e.g. 2 cos x = 1)	Isolate trig ratio, use inverse, find all in range	”Solve cos x = 0.5”
Quadratic in sin/cos	Let u = sin x, solve u² − u = 0, reject |u| > 1	”Solve 2 sin² x − sin x = 0”
tan x = k	One solution per 180° period	”Solve tan x = −1, 0° ≤ x ≤ 360°“
Graphical	Read intersections from a sketch	”Use the graph to solve sin x = 0.5”

Trigonometric Equations in past-paper wording: command words that matter

Most lost marks come from misreading the range or stopping after the calculator’s first answer. These are the command words you will see.

Command word / phrase	What the question wants	Typical trig stem
Solve	Find every value in the stated interval	”Solve 3 sin x + 1 = 0 for 0° ≤ x ≤ 360°.”
Write down	State value(s); working may be brief	”Write down the two values of x.”
Show that	Prove x = … is a solution — substitute back	”Show that x = 30° is a solution of …”
Give your answers correct to …	Round angles as instructed	”Give your answers correct to 1 decimal place.”
Give your answers in radians	Express in terms of π where possible	”Give your answers in radians, 0 ≤ x ≤ 2π.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the algebra — is what method marks reward.

1. “Solve sin x = 0.5 for 0° ≤ x ≤ 360°.” x = sin⁻¹(0.5) = 30°. Sin is also positive in quadrant II: 180° − 30° = 150°. Mark-scheme reward: both angles listed.
2. “Solve 2 cos² x − cos x − 1 = 0 for 0° ≤ x ≤ 360°.” Let u = cos x: (2u + 1)(u − 1) = 0 → u = −½ or u = 1. cos x = −½ → 120°, 240°; cos x = 1 → 0°, 360°. Reward: factorisation shown, invalid cos values rejected.
3. “Solve tan x = −1 for 0° ≤ x ≤ 360°.” x = tan⁻¹(−1) = 135°; next period: 135° + 180° = 315°. Reward: both solutions in range.

When you can recognise the wording instantly, work the full set on the Trigonometry topical past paper questions and the Trigonometric Equations quiz to lock the method in.

How Trigonometric Equations connect to the rest of Trigonometry

Trig equations depend on Trigonometric Graphs for symmetry and on Right Angled Trigonometry for ratio recall. They also feed into 3D Trigonometry when angles must be found before lengths. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Giving only the calculator’s first answer when the question asks for all solutions in 0° ≤ x ≤ 360°.
• Forgetting to reject impossible values — e.g. sin x = 1.5 has no real solution.
• Using DEG mode when the question specifies radians (or vice versa).
• In quadratic trig equations, not factorising before trying to use inverse sin/cos.
• Rounding the principal value too early, so the second solution is wrong.

When you need more support

If trig equations keep tripping you up — especially quadratic types or radian answers — work through the Trigonometry topical past paper questions and the Trigonometric Equations quiz to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Are Trigonometric Equations hard in Cambridge IGCSE Maths? No — the algebra is straightforward. Marks are lost when students give only one solution, use the wrong angle mode, or forget to reject values where |sin x| > 1.

How do I find all solutions between 0° and 360°? Find the principal value with the inverse function, then use the CAST diagram or graph symmetry to write the other angle(s) in the same interval.

What if the equation has sin² x or cos² x? Treat it as a quadratic: substitute u = sin x or u = cos x, factorise, solve for u, then find x — and reject any u outside [−1, 1].

How do I revise Trigonometric Equations effectively? Read the subtopic notes, practise listing all solutions on every question, then take the Trigonometric Equations quiz. Revisit CAST and graphs if second solutions keep going wrong.

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